Recognition Methods of Geometrical Images of Automata Models of Systems in Control Problem

The laws of functioning of discrete deterministic dynamical systems are investigated, presented in the form of automata models defined by geometric images. Due to the use of the apparatus of geometric images of automata, developed by V.A. Tverdokhlebov, the analysis of automata models is carried out on the basis of the analysis of mathematical structures represented by geometric curves and numerical sequences. The purpose of present research is to further develop the mathematical apparatus of geometric images of automaton models of systems, including the development of new methods for recognizing automata by their geometric images, given both geometric curves and numerical sequences.

Effect of a Mode of Update on Universality Class for Coupled Logistic Maps: Directed Ising to Ising Class

Ising model at zero temperature leads to a ferromagnetic state asymptotically. There are two such possible states linked by symmetry, and Glauber–Ising dynamics are employed to reach them. In some stochastic or deterministic dynamical systems, the same absorbing state with [Formula: see text] symmetry is reached. This transition often belongs to the directed Ising (DI) class where dynamic exponents and persistence exponent are different. In asymmetrically coupled sequentially updated logistic maps, the transition belongs to the DI class. We study changes in the nature of transition with an update scheme. Even with the synchronous update, the transition still belongs to the DI class. We also study a synchronous probabilistic update scheme in which each site is updated with the probability [Formula: see text]. The order parameter decays with an exponent [Formula: see text] in this scheme. Nevertheless, the dynamic exponent [Formula: see text] is less than [Formula: see text] even for small values of [Formula: see text] indicating a very slow crossover to the Ising class. However, with a random asynchronous update, we recover [Formula: see text]. In the presence of feedback, synchronous update leads to a transition in the DI universality class which changes to Ising class for synchronous probabilistic update.

Dynamical system approaches to climate variability

A tutorial is provided on the application of dynamical systems theory to problems in climate dynamics. We start with the analysis of low-dimensional deterministic dynamical systems using bifurcation theory and provide examples in conceptual climate models.We then proceed to stochastic low-dimensional systems and eventually end with operator-based techniques within ergodic theory. In these notes, we start each section from a climate dynamics problem, motivate the choice of the model to study it, and use dynamical systems analysis to understand the behavior of the model solutions. In each of the chapters, a different phenomenon, a different type of model, and/or a different dynamical system tool will be presented.

Rates in almost sure invariance principle for slowly mixing dynamical systems

We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau–Manneville intermittent maps, with Hölder continuous observables. Our rates have form $o(n^{\unicode[STIX]{x1D6FE}}L(n))$, where $L(n)$ is a slowly varying function and $\unicode[STIX]{x1D6FE}$ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed $O(n^{1/4})$. To break the $O(n^{1/4})$ barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes–Liu–Wu and Cuny–Dedecker–Merlevède on the Komlós–Major–Tusnády approximation for dependent processes.

New geography-mathematic approaches in the tasks of design of distribution of harmful admixtures in an atmosphere

It is presented a qualitative overview of the new conceptual approaches, which are based on the provisions of the chaos theory, dynamical systems theory, fractal geometry, analysis of Lyapunov exponents, and others, to problems of modeling the propagation of pollution impurities in the atmosphere of industrial cities and predicting the evolutionary dynamics. We summarize the main ideas of these approaches with emphasis on the analysis of time series of concentrations of pollution impurities in the atmosphere, as well as an analysis that shows that the chaotic regime of the time evolution of the characteristics of deterministic dynamical systems, in particular, the application of ecological systems is, in fact, a non-linear phenomenon which in principle can not be described on the basis of the classical linear regular-dynamic models.